
Since expectation values of the identity operator are equal to 1, i.e. \(\langle \phi  |I^{\otimes n} | \phi  \rangle = \langle \phi ^{\prime } | I^{\otimes n} | \phi ^{\prime } \rangle  = 1\) , evaluating the expectation value of the operator \(A_{x, \beta }\)  requires only the evaluation of expectation values of the simple Hamiltonians \(H_{1-4}\)  (\(H_{1-3}\)  for Dirichlet boundary condition). The required number of quantum circuits is therefore limited to a constant \(O(n^0)\)  [1]}. Similar decomposition expressions apply to problems of higher dimensions, including \(A_{y,\beta }\)  in the \(y\)  direction [2]}.
